Question 1: The sum of two numbers is and their difference is . Find the numbers.

__Answer:__

Let the two numbers be

Add (i) and (ii)

Therefore

Hence the two numbers are

Question 2: Twice a number is equal to thrice the other number. If the sum of the numbers is , find the numbers.

__Answer:__

Let the two numbers are

Solving for

or

Therefore

Hence the two numbers are .

Question 3: Find two numbers such that twice the first added to thrice the second gives and twice the second added to thrice the first gives .

__Answer:__

Let the numbers be

Solving for

Multiply i) by 2 and ii) by 3 and subtract ii) from i)

Substituting in i)

Hence the two numbers are

Question 4: Find two numbers which differ by and are such that four times the larger added to three times the smaller gives .

__Answer:__

Let the numbers be

Solving by

Multiply i) by 4 and subtract ii) from i)

Calculating for

Hence the two numbers are

Question 5: The sum of two numbers is and the difference of their squares is . Find the numbers.

__Answer:__

Let the two numbers be

Simplifying ii)

Solving for

Add i) and iii)

Substituting in i)

Hence the numbers are

Question 6: The sum of the digits of a two-digit number is . On adding to the number, its digits are reversed. Find the number.

__Answer:__

Let the two digit numbers be

Simplifying ii)

Solving for

Add i) and iii)

Substituting in i)

Hence the numbers is

Question 7: Two digit number is three times the sum of its digits. lf is added to the number, its digits are reversed. Find the original number.

__Answer:__

Let the two digit numbers be

Simplifying i) and ii)

Solving for

Multiplying iv) by 7 and Subtract iv) from ii)

Hence

Therefore the numbers is

Question 8: A two-digit number is seven times the sum of its digits. If is subtracted from the number, its digits get interchanged. Find the number.

__Answer:__

Let the two digit number be

Solving for

Multiplying ii) by 3 and Subtract ii) from i)

Hence

Therefore the numbers is

Question 9: Find a fraction which reduces to when is added to both its numerator and denominator; and reduces to when is added to both its numerator and denominator.

__Answer:__

Let the fraction be

Solving for

Multiplying i) by 5 and ii) by 3 and subtract ii) from i)

Hence

Therefore the fraction is

Question 10: On adding to the numerator of a fraction, it becomes . Also, on adding to the denominator of the original fraction, it becomes . Find the original fraction.

__Answer:__

Let the fraction be

Solving for

Multiplying i) by 3 and ii) by 2 and subtract ii) from i)

Hence

Therefore the fraction is

Question 11: In a given fraction, if the numerator is multiplied by and the denominator is reduced by , we get . But, if the numerator of the given fraction is increased by and the denominator is doubled, we get . Find the fraction.

__Answer:__

Let the fraction be

Solving for

Multiplying ii) by 2 and subtract ii) from i

Hence

Therefore the fraction in

Question 12: years ago, a lady was thrice as old as her daughter. years hence, the lady would be twice as old as her daughter. What are their present ages?

__Answer:__

Let the present age of lady

Let the Present age of Daughter

Solving for

Subtract ii) from i)

Hence

Therefore :

Lady’s Age

Daughter’s Age

Question 13: The sum of the ages of A and B is years. In years’ time, the age of A will be twice the age of B. Find their present ages.

__Answer:__

Let A’s Age

Let B’s Age

Solving for

Subtract ii) from i)

or

Therefore

A’s Age

B’s Age

Question 14: A is years elder than B. years ago A was four times as old as B. Find their present ages.

__Answer:__

Let the age of B

Hence A’s Age

Therefore

B’s Age

A’s Age

Question 15: Six years ago, the ages of Geeta and Seema were in the ratio . Nine years hence, their ages will be in the ratio . Find their present ages.

__Answer:__

Let the age of Geeta

Let the age of Seema

Solving for

Multiplying i) by 7 and ii) by 4 and Subtract ii) from i)

Hence

Therefore

Geeta’s Age

Seema’s Age

Question 16: knives and forks cost Rs. , while knives and, forks together cost Rs. . Find the cost of a knife and that of a fork.

__Answer:__

Let the cost of knives

Let the Cost of fork

Solving for

Multiplying i) by 3 and ii) 2

Hence

Hence Cost of :

Knive

Fork

Question 17: The cost of cups and spoons is Rs. , while the cost of cups and spoons is Rs. . Find the cost of cups and spoons.

__Answer:__

Let the cost of Cup

Let the Cost of Spoon

Solve for

Multiply i) by 16 and ii) by 13 and subtract ii) from i)

Hence

Hence Cost of:

Cup

Spoon

Question 18: Rahul covered a distance of km in hours, partly on bicycle at kmph and partly on moped at kmph. How much distance did he cover on moped?

__Answer:__

Total Distance

Total Time

Let the distance covered by cycle

Let the Distance covered by moped

Simplify

Hence Distance Covered by:

Cycle

Moped

Question 19: A boat can go km downstream in hours and, km upstream in hours. Find i) the speed of the boat in still water (ii) the rate of the current.

__Answer:__

Let the Speed of boat in still water

Speed of Stream

Solve for , add i) and ii)

Speed of Stream

Question 20: The monthly incomes of A and B are in the ratio and their monthly savings are in the ratio . If each spends Rs. per month, find the monthly income of each.

__Answer:__

Let A’s Income

Let B’s Income

Substituting

Hence

Therefore :

A’s Income

B’s Income

Question 21: nuts and bolts weigh grams, while nuts and bolts weigh grams. Find the weight of each nut and that of each bolt. How much do nuts and bolts weigh together?

__Answer:__

Let the Weight of Nut

Let the Height of Bolt

Solving for

Multiply i) by 8 and ii) by 6 and Subtract ii) from i)

Hence

Weight of:

Nut

Bolt

Therefore 3 Nuts and 3 Bolts will weight

Question 22: There are some girls in two classrooms, A and B. If girls are sent from room A to room B, the number of girls in both the rooms will become equal. If girls are sent from room B to room A, then the number of girls in room A would be double the number of girls in room B. How many girls are there in each class room?

__Answer:__

Let No. of girls in classroom

Solving ,

Subtract ii) from i)

Hence

No. of Girls in Class Room:

Question 23: men and boys can do a piece of work in days, while men and boys can finish it in days. How long would it take man alone to do it?

__Answer:__

Suppose 1 man finished the work in days and 1 Boy finished the work in days

There 1 man’s 1 day’s

And, 1 Boy’s 1 Day’s work

Now 4 men and 4 Boys finished the work in 3 days

Similarly 2 men and 5 boys finished in 4 days

Solving for

Multiply ii) by 2 and Subtract ii) from 2

or

Similarly

So one men can finished the work in

Question 24: A takes hours longer than B to walk km. But, if A doubles his pace, he is ahead of B by hour minutes. Find the speeds of A and B.

__Answer:__

Let the Speed of A

Let the Speed of B

Time taken By A

Time Taken By B

If A Double him Race Time taken by A

Therefore

Hence

Question 25: If the length of a rectangle is reduced by m and breadth increased by m, its area increases by m^{2}. If however, the length is increased by m and breadth reduced by m, then the area is reduced by m^{2}. Find the length and breadth of the rectangle.

__Answer:__

Let the Length of the rectangle

Let the Breadth of the rectangle

Solving for

From i)

Substituting in ii)

Hence

Therefore:

Length

Breadth

Sir, may I know the font you use.

Love that font 🙂

Cambria Math